1. IAG Scientific Assembly – Cairns, Australia, 22-26 August 2005 The GOCE Mission GOCE (Gravity field and steady-state Ocean Circulation Explorer) will be the first satellite gradiometry mission. It has been designed for the determination of the stationary gravity field with high accuracy and spatial resolution. GOCE will be continuously tracked by the GPS system. The on-board three-axis gradiometer will measure the second order derivatives of the gravitational potential, the so called gradiometric observations. To analyze this new type of observations, three different approaches have been proposed, one of which is the space-wise approach. The Space-Wise Approach The Wiener Filter Updating the observations Harmonic analysis Conclusions The End to End simulation presented here has been very useful both to test the software and the method, and to obtain improvements or identify any that have to be made. The baseline solution can meet the GOCE requirement of solving the gravity field spherical harmonic expansion up to degree 200. Some aspects of the method (e.g. optimal size of gridding window) cannot be tested using only one month of data The real mission will cover a time span of 12 months. Exploiting the spatial correlation of all these observations is expected to improve the results significantly, both in terms of accuracy and spatial resolution. GOCE: a full-gradient solution in the space-wise approach Federica Migliaccio – Mirko Reguzzoni – Fernando Sansò – Nikolaos Tselfes Federica Migliaccio – Mirko Reguzzoni – Fernando Sansò – Nikolaos TselfesPolitecnico di Milano - Italy Carl Christian Tscherning – Martin Veicherts Carl Christian Tscherning – Martin VeichertsUniversity of Copenhagen – Denmark According to the space-wise approach, the gravity field model is estimated by solving a boundary value problem for a “sphere” at the satellite altitude. Using the Satellite to Satellite Tracking data (SST), the potential is estimated via the Energy Conservation method. The spectra of the potential and the measured Satellite Gravity Gradients (SGG) are computed by Fast Fourier Transform (FFT) and the Wiener Filter (WF) is applied to them. The filtered data are transformed back to the space domain, and are used for the estimation of gridded values at mean satellite altitude. Finally, applying the Harmonic Analysis operator to these gridded data, the coefficients of a spherical harmonic model are estimated. The procedure is then iterated in order to improve the along-orbit signal estimates, using the so-called Complementary Wiener Filter and Rotation Correction. The performance of this space-wise scheme has been tested, on the basis of a realistic simulated data set. The Simulated Data The test data-set has been provided by ESA (European Space Agency). It consists of observations spanning the duration of one month, at 1 sec sampling rate. The gravity gradients are based on the EGM96 model, up to degree and order 360, and were contaminated with heavily coloured noise. The satellite orbit, including positions, velocities, accelerations and attitude quaternions, was simulated based on EGM96 up to degree and order 50. The energy conservation was not used here, so the potential along the orbit was directly generated (up to degree 360), including white noise of σ = 0.3 m 2 /s 2 The axes of the gradiometer (GRF) (x,y,z) are not coincident with the local orbital reference frame (LORF) (along-track: , cross-track: , radial: r). The latter is the reference system in which the filtering is made. The direct rotation of the observed gravity-gradient tensor cannot be applied, since it would spread the large errors of the off-diagonal components onto all the tensor components. Therefore a first prediction of the diagonal components in the LORF frame is performed by neglecting the off-diagonal observations. The missing rotations terms are then corrected iteratively. The gradients in LORF, and the potential, are transformed to the frequency domain via FFT, and a Wiener Filter (WF) is applied to them. Data gridding FFT complementary Wiener filter Data synthesis along orbit Wiener filter FFT + SST + Energy conservation FFT LORF/GRF correction test SGG Harmonic analysis Space-wise solver Final model Future Work Power Spectral Densities (PSD) of the signal (blue) and noise (red). The gradiometer noise is not stationary (e.g. “peaks” at low frequencies). The energy conservation method will be used in future tests with common-mode acceleration data. A new version of the Wiener filter will be used, that is expected to give rise to even more satisfying results, also for the T  and T  r component A statistical homogenisation of the observed gravity field prior to the gridding procedure will be tested. Error estimates of both Wiener filter and gridding have to be better tuned. Parameters used in this simulation: undersampling = 5 sec rate interpolation area = 10°  10° (2° overlapping) final grid size = 0.72°  0.72° r.m.sT nn [mE]T rr [mE]T [m 2 /s 2 ] Gridding3.048.410.0675 The synthesis of the observations along the orbit is made using the coefficients of the latest computed model to: recover the signal lost due to the Wiener filtering, especially at low frequencies, by applying a complementary filter to the synthesised observations. estimate the ignored rotation terms between GRF and LORF, due to the noisy off-diagonal components. These two correction terms, added to the filtered data, result in a significant decrease of the estimation error along the orbit. Then the gridding and the harmonic analysis are repeated. The iteration is terminated at convergence. r.m.s.T  [mE]T  [mE]T rr [mE]T [m 2 /s 2 ] iteration 04.36924.4289.9680.088 iteration 11.33310.4473.6850.054 iteration 21.32310.4063.6180.054 GEOCOL T  T rr T nn (grid) ^ ^ ^ T rr (grid) ^ GEOCOL T ^ T (grid) ^ T  ^ Gravity anomaly errors (global r.m.s. 5 mgal) r.m.s.T  [mE]T  [mE]T rr [mE]T [m 2 /s 2 ] Before WF269.2246.8548.00.299 After WF4.36924.4289.9680.088 The gridded values in a local East-North-Radial (e,n,r) reference frame are computed by least squares collocation applied to regional patches of filtered data. Two methods are possible for the harmonic analysis The Fast Spherical Collocation (FSC), which assimilates in a statistical mode a priori knowledge on the field, in terms of prior degree variances. This is the baseline solution (and used in the next iterations). The INTegration method (INT), which exploits the orthogonality of the spherical harmonics. This is used as a check solution. The FSC method works better at low degrees, while INT at high degrees. The improvement between iterations 0 and 1 is significant. The improvement at iteration 2 is very small, meaning that there is a fast convergence. Two or three iterations are enough. The gravity anomaly errors are larger in areas where the field is less smooth, such as the Himalayas or the Andes. The estimation errors become smaller than the along-track errors, especially at high latitudes, where the data are denser. The predicted errors are about one order of magnitude smaller than the real ones. Due to the non stationarity of the noise and the correlation between the noise and the signal, the data are not filtered all together in a 4 dimensional Wiener Filter (WF 4D ). The potential and the T rr gradient are jointly filtered (WF 2D ), while the other two components are filtered separately (WF 1D ). T rr TT ^ ^ WF 2D The mean square error of the data before and after WF The output of the WF are data streams with much less noise, and a reasonable prediction of the covariance of the estimation error along the orbit, to be used in the gridding procedure. Error covariance of T rr after WF empirical estimation error T rr estimated gridded dataT rr estimated errors T  ^ WF 1D T  ^ WF 1D PSD of the potentialPSD of T xx PSD of T yy PSD of T zz Empirical (blue) Predicted (red) Modelled (green) T nn (grid) T rr (grid) T (grid) SH coeff. error estimate ^ ^ ^ FSC ^ T (grid) T rr (grid) T (grid) ^ ^ SH coeff. error estimate INT Coefficient relative error The Gridding Along-orbit estimation error at iterations 0,1 and 2 Acknowledgements This work has been prepared under ESA contract 18308/04/NL/NM (GOCE High-level Processing Facility). Hz sec mE 2 mE mgal EGM96 Error degree variances of the estimated coefficients INT FSC EGM96 Error degree variances at iterations (0) and (1) INT (0) FSC (0) INT (1) FSC (1) m 2 /s 2 Hz E Hz